curve #944

y² + y = x³ + x² + x − 1

a-invariants: [0, 1, 1, 1, -1]
rank (lower bound): ≥ 0
torsion subgroup: ℤ/3ℤ
conductor (N): 51
discriminant (Δ): -459
Faltings height: -0.8085
naive height: 13.5416
primes of bad reduction: 3, 17
regulator: 1
submitted by: David Renshaw
submitted at: 2026-09-23 15:04:39 UTC
last updated: 2026-09-23 15:04:39 UTC

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Commentary

Cremona 51a1 = LMFDB 51.a2: https://www.lmfdb.org/EllipticCurve/Q/51/a/2 Mordell–Weil rank exactly 0 (Cremona's tables). Conductor 51, minimal discriminant -459, torsion ℤ/3ℤ.

last edited by David Renshaw at 2026-09-23 15:04:39 UTC · history